How do you define the "boundary" of a topological space?

The notion of boundary that you are looking for comes from the definition of topological manifolds with boundary. As opposed to a regular manifold $X$, a manifold with boundary has the property that each point in $X$ has an open neighborhood which is homeomorphic to an open set in the euclidean half space $\mathbb{R}_+^n=\{(x_1,\dots,x_n)\in\mathbb{R^n}:x_n\ge0\}$. Thus we then define $\partial X$ to be the points which when mapped to $\mathbb{R}_+^n$ have $x_n=0$.

This definition has the benefit that an embedding of $X$ into some other space does not change $\partial X$. Thus $\partial D^2=S^1$ irregardless of whether you view it as living in $\mathbb{R}^2$ or in $\mathbb{R}^3$.

The concept of boundary can be extended to the (regular) CW-complexes, as suggested here.

The boundary of a (regular) CW-complex $X$ is : $$\partial X := \overline{\bigcup_{n \ge 0}(\bigcup_{c \in \text{n-cells}} \partial c) / (\bigcup_{c \ne c'\in \text{n-cells}} (\partial c \cap \partial c'))}$$

Definition : The notation "$n$-cells" above, is the set of closed $n$-cells.

Example : Let $X$ be a topological space with the following simplicial complex structure :
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All the sets :

$0$-cells $=\{ A,B,C,D,E,F \}$
$1$-cells $=\{ [A,B],[B,C],[C,D], [D,E],[E,A],[A,F]... \}$
$2$-cells $=\{ [A,B,F],[B,C,F],[C,D,F], [D,E,F],[E,A,F]\}$

Now :

$\partial A = \partial B = ... = \partial F = \emptyset$
$\partial [A,B] = \{A,B \}$ , $\partial [B,C] = \{B,C \}$ , ....
$\partial [A,B,F] = [A,B] \cup [B,F] \cup [A,F] $, $\partial [B,C,F] = [B,C] \cup [C,F] \cup [B,F] $, ...

So :

$(\bigcup_{c \in \text{0-cells}} \partial c) / (\bigcup_{c \ne c'\in \text{0-cells}} (\partial c \cap \partial c')) = \emptyset$
$(\bigcup_{c \in \text{1-cells}} \partial c) / (\bigcup_{c \ne c'\in \text{1-cells}} (\partial c \cap \partial c')) = \emptyset$
$(\bigcup_{c \in \text{2-cells}} \partial c) / (\bigcup_{c \ne c'\in \text{2-cells}} (\partial c \cap \partial c')) = (A,B) \cup (B,C) \cup (C,D) \cup (D,E) \cup (E,A)$

Conclusion : $\partial X = [A,B] \cup [B,C] \cup [C,D] \cup [D,E] \cup [E,A]$

Questions : Let $X$ be a topological space admitting a (regular) CW-complex structure :